## Abstract

Sibly *et al*. (Reports, 22 July 2005, p. 607) recently estimated the relationship between population size and growth rate for 1780 time series of various species. I explain why some aspects of their analysis are questionable and, therefore, why their results and estimation procedure should be used with care.

My concern with the results of Sibly *et al*. (*1*) revolves around the interpretation of their parameters. They model the per capita growth rate (*pgr*) of populations by the theta-logistic equation (1) where *N* is the population size, *r*_{0} is a parameter representing *pgr* when *N* = 0, *K* is the carrying capacity of the population, and θ is a parameter describing the curvature of the relationship. When *N* = 0, the population growth rate is 0 and, therefore, *r*_{0} should be defined as the rate of population growth at low densities, as Reynolds and Freckleton (*2*) did in their discussion of Sibly *et al*. (*1*), or perhaps even more accurately as the *pgr* of the population in the absence of density-dependent regulation. Although this clarification may appear trivial, its importance will become apparent. Also, the parameter θ should be defined to take values (strictly) greater than 0. Sibly *et al*. (*1*)allow θ to be negative and 0. In fact, they estimate θ to be negative for real populations [see figure 1C in (*1*)]. The reason θ should be greater than 0 can be seen by considering θ negative in Eq. 1. In this case, we have (2) Thus, provided *r*_{0} is positive (which appears necessary from the above discussion), when the population is below its carrying capacity, *pgr* is negative, resulting in population extinction. Further, when the population is above its carrying capacity, *pgr* is positive, resulting in unbounded growth. Sibly *et al*. (*1*) allow *r*_{0} to be negative. This can be seen from consideration of their analysis of the insect population *Xylena vetusta* [GPDD ID 6321 (*3*)], presented in figure 1C in (*1*). They estimate that *K* = 512 and θ = –2.0 for this population. From the figure, we can see that when *N* is approximately 1200, *pgr* is approximately –0.5. Thus, using Eq. 1, *r*_{0}≈ –0.61. Therefore, if θ is allowed to take negative values, *r*_{0} must also be negative, so its physical interpretation is lost.

Sibly *et al*. (*1*), in their Supporting Online Material (SOM), discuss two mathematical oddities in Eq. 1. They first point out that specifying the value of *pgr* for low *N* at *N* = 0 or at *N* > 0 makes a big difference, as seen by comparing their figure 2 and figure S1 when θ = 0. In figure 2 in (*1*), the curves are said to be constrained to go through (*N, pgr*) equaling (1, 0.1), and in figure S1 (0, 0.1) (a seemingly impossible scenario when θ ≤ 0). When θ is greater than 0, the curves are almost identical. The big differences occur when the parameter θ takes negative values (and the value 0). However, this is not due to a “mathematical oddity” but is simply because *pgr* → –∞ as *N* → 0 when θ < 0 and *r*_{0} > 0, and *pgr* is finite when *N* = 1 [thus allowing Sibly *et al*. to choose *r*_{0}(< 0) such that *pgr* = 0.1]. In the case θ = 0, we need to consider the “second mathematical oddity.” Sibly *et al*. (*1*) state, “It turns out that when θ → 0, *r*_{0} → ∞ in such a way that *r*_{0}θ assumes a finite value not equal to 0.” This cannot simply “turn out”; when θ = 0, Eq. 1 is clearly equal to 0 for all *N*, independent of the value of *r*_{0}, and thus the requirement that the above limiting process takes place is clearly an assumption of the authors'.

My concern with the method of estimation used by Sibly *et al*. is as follows. They state that they fitted Eq. 1 to each of the 3296 tractable time series in the GPDD using a nonlinear least-squares procedure. However, the method for estimating θ described in the SOM does not use a nonlinear least-squares procedure. Instead, Sibly *et al*. fix θ at various values (including negative values), regress log_{e}(*N*_{t+1}/*N*_{t}) on *N*^{θ}_{t}, and choose the value that gives the lowest residual sum of squares. The constant of this linear regression corresponds to *r*_{0}, and the coefficient of *N*^{θ}_{t} corresponds to –*r*_{0}/*K*^{θ}. Although this procedure may result in a model that provides a good fit to the data [keeping in mind that log_{e}(*N*_{t+1}/*N*_{t}) is only a proxy for *pgr*], this estimated model may not be sensible for all population sizes, and *r*_{0} may have no physical interpretation. Considering figure 1C in (*1*), we can see that the model explains the data well, but this model predicts that *pgr* is approximately 6396 when *N* = 5, and approximately 159,907 when *N* = 1, a questionable result, and, as already discussed, *r*_{0}≈ –0.61.

Although I do not believe the above concerns have had a major effect on the overall conclusions of Sibly *et al*. (*1*), I do believe they raise an important issue: whether we wish to model the relationship between abundance and growth only for population sizes contained in our data set [again keeping in mind that log_{e}(*N*_{t+1}/*N*_{t}) is just a proxy for *pgr*], and thus *r*_{0} may possibly have no physical interpretation, or to have a relationship that is sensible for all *N* and thus retains the physical interpretation of *r*_{0}. The latter can only be achieved with θ > 0. In this case, a reasonable fit to the data should still be achievable [cf., Fig. 1 below with figure 2 from (*1*)]. If this is not the case, then the user should perhaps consider a model other than Eq. 1 to explain the relationship between abundance and growth for the population in question.